Examine the sign of u(n+1) - u_n

The difference method

To investigate the direction of variation of a sequence (u_n), the most general method is to calculate and simplify the expression D = u(n+1) - u_n, and then to determine its sign for every integer n.

• If D ≥ 0 for all n, the sequence is increasing (D > 0: strictly increasing).
• If D ≤ 0 for all n, the sequence is decreasing (D < 0: strictly decreasing).
• If D = 0 for all n, the sequence is constant.

Example 1: arithmetic sequence

Let $u_n = u_0 + n\cdot r$. Then $u(n+1) - u_n = r$, which is a constant. The sign of $r$ is therefore sufficient to conclude:

Sign of $r$	Direction of variation
r > 0	strictly increasing
r < 0	strictly decreasing
r = 0	constant

Example 2: non-affine sequence

Let u_n = n² - n. We calculate: u(n+1) - u_n = (n+1)^2 - (n+1) - (n^2 - n) = n^2 + 2n + 1 - n - 1 - n^2 + n = 2n.

Since n is a natural number, 2n ≥ 0 for all n, so the sequence is non-decreasing in the broad sense (and strictly non-decreasing once n ≥ 1, as 2n > 0 for n ≥ 1).

A common pitfall

A frequent mistake is to draw conclusions based on the sign of $u_n$ itself rather than on the difference $u(n+1) - u_n$. The sign of the terms of the sequence has no direct bearing on its direction of variation: a sequence may have all its terms positive and yet be decreasing (for example, u_n = 1/(n+1)).